A wood thrush is a bird common in the Northeastern part of the United States; it is known for its "loud clear song." In the poem, Eliot refers to hearing the wood thrush "singing through the fog". He mentions this in the opening of the poem, and again at the closing of the poem.
The mentioning of the woodthrush can serve several purposes. One is to set the scene, to use imagery (the 5 senses) to create a mood and help the reader to feel like they are right there, where the poet is describing. The entire opening and closing stanzas set the scene through sights and sounds ("What seas what shores what grey rocks and what islands/ What water lapping the bow /And scent of pin"), and he puts the woodthrush in the fog in there too. Now the reader can feel like they are actually there, in body. It also sets a mood; he's in a boat, it is foggy, and you can hear the singing of a bird somewhere close by.
The second purpose in mentioning the woodthrush is for structure and closure. He opens and closes with a repeated mentioning of it; this helps "sandwich" the poem, and gives it a nice, tidy, closure, almost like the beginning and ending of a story.
A third reason for the mentioning of the wood thrush is symbolism. It is known for its loud and clear song. Note that Eliot put it in a fog; a fog is confusing and disorienting, but the woodthrush's song was heard clearly through it, piercing through the fog. This could symbolize the clarity that Eliot is having at that moment, in regards to death. The entire poem discusses death and its impact. It discusses his confusion and lack of clarity in the past and how he wants to "resign my life for this life". So, as he sits there, in that spot and time, in that fog, he has a moment of clear, calm thinking where his eyes are opened, and that could be symbolized by the thrush's song through the murky fog.
I hope that helps a bit! Good luck!